3.912 \(\int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx\)

Optimal. Leaf size=48 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\sqrt {a+b^2 x^4}-b x^2}}\right )}{\sqrt {2} \sqrt {b}} \]

[Out]

1/2*arctan(x*2^(1/2)*b^(1/2)/(-b*x^2+(b^2*x^4+a)^(1/2))^(1/2))*2^(1/2)/b^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2132, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\sqrt {a+b^2 x^4}-b x^2}}\right )}{\sqrt {2} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]

[Out]

ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 2132

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+2 b x^2} \, dx,x,\frac {x}{\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}\right )}{\sqrt {2} \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 48, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\sqrt {a+b^2 x^4}-b x^2}}\right )}{\sqrt {2} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]

[Out]

ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])

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fricas [A]  time = 1.80, size = 146, normalized size = 3.04 \[ \left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{b}} \log \left (4 \, b^{2} x^{4} - 4 \, \sqrt {b^{2} x^{4} + a} b x^{2} + 2 \, {\left (\sqrt {2} b^{2} x^{3} \sqrt {-\frac {1}{b}} - \sqrt {2} \sqrt {b^{2} x^{4} + a} b x \sqrt {-\frac {1}{b}}\right )} \sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}} + a\right ), -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}}}{2 \, \sqrt {b} x}\right )}{2 \, \sqrt {b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

[1/4*sqrt(2)*sqrt(-1/b)*log(4*b^2*x^4 - 4*sqrt(b^2*x^4 + a)*b*x^2 + 2*(sqrt(2)*b^2*x^3*sqrt(-1/b) - sqrt(2)*sq
rt(b^2*x^4 + a)*b*x*sqrt(-1/b))*sqrt(-b*x^2 + sqrt(b^2*x^4 + a)) + a), -1/2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-b
*x^2 + sqrt(b^2*x^4 + a))/(sqrt(b)*x))/sqrt(b)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}}}{\sqrt {b^{2} x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-b \,x^{2}+\sqrt {b^{2} x^{4}+a}}}{\sqrt {b^{2} x^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x)

[Out]

int((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}}}{\sqrt {b^{2} x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\sqrt {b^2\,x^4+a}-b\,x^2}}{\sqrt {b^2\,x^4+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b^2*x^4)^(1/2) - b*x^2)^(1/2)/(a + b^2*x^4)^(1/2),x)

[Out]

int(((a + b^2*x^4)^(1/2) - b*x^2)^(1/2)/(a + b^2*x^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- b x^{2} + \sqrt {a + b^{2} x^{4}}}}{\sqrt {a + b^{2} x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**2+(b**2*x**4+a)**(1/2))**(1/2)/(b**2*x**4+a)**(1/2),x)

[Out]

Integral(sqrt(-b*x**2 + sqrt(a + b**2*x**4))/sqrt(a + b**2*x**4), x)

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